Utility maximisation and utility indifference price for exponential semi-martingale models and HARA utilities
The authors consider the utility maximisation problem for semi-martingale models and HARA (hyperbolic absolute risk aversion) utilities. The market is assumed to consist of one non-risky and two risky assets, possibly with jumps, represented as the stochastic exponents of two corresponding processes, \(X^{(1)}\) and \(X^{(2)}\). Process \(X^{(1)}\) us supposed to satisfy the Blumenthal--Getoor law at zero and possess predictable representation property. Corresponding market is supposed to be incomplete. The investor can trade the first asset until some maturity date and also has a European--type contingent claim for the second one, with the bigger maturity date. It may happen that the second asset can not be traded therefore the investor would like to evaluate the option for this asset. The price of the option in such a situation can be assessed by its indifferent price. So, it is possible to formulate the utility maximization problem in a natural setting. Using specific properties of HARA utilities, the authors reduce the initial maximization problem to the conditional one, which is solved by applying a dual approach. Then the solution of the conditional maximization problem is expressed via conditional information quantities related to HARA utilities, like the Kullback-Leibler information and Hellinger-type integrals. In turn, the information quantities are expressed in terms of information processes, which is helpful in indifference price calculus. Finally, equations for indifference prices are given. The results are applied to Black-Scholes models with correlated Brownian motions.
scientific article; zbMATH DE number 6473880
scientific article